Theorem cdleme31sdnN 34693

Description: Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 31-Mar-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cdleme31sdn.c	⊢ 𝐶 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊)))
cdleme31sdn.d	⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
cdleme31sdn.n	⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐶)

Assertion

Ref	Expression
cdleme31sdnN	⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, ⦋𝑠 / 𝑡⦌𝐷)

Distinct variable groups:   𝑡, ∨   𝑡, ∧   𝑡,𝑃   𝑡,𝑄   𝑡,𝑈   𝑡,𝑊   𝑡,𝑠

Allowed substitution hints:   𝐶(𝑡,𝑠)   𝐷(𝑡,𝑠)   𝑃(𝑠)   𝑄(𝑠)   𝑈(𝑠)   𝐼(𝑡,𝑠)   ∨ (𝑠)   ≤ (𝑡,𝑠)   ∧ (𝑠)   𝑁(𝑡,𝑠)   𝑊(𝑠)

Proof of Theorem cdleme31sdnN

Step	Hyp	Ref	Expression
1		cdleme31sdn.n	. 2 ⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐶)
2		biid 250	. . 3 ⊢ (𝑠 ≤ (𝑃 ∨ 𝑄) ↔ 𝑠 ≤ (𝑃 ∨ 𝑄))
3		vex 3176	. . . 4 ⊢ 𝑠 ∈ V
4		cdleme31sdn.d	. . . . 5 ⊢ 𝐷 = ((𝑡 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑡) ∧ 𝑊)))
5		cdleme31sdn.c	. . . . 5 ⊢ 𝐶 = ((𝑠 ∨ 𝑈) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑠) ∧ 𝑊)))
6	4, 5	cdleme31sc 34690	. . . 4 ⊢ (𝑠 ∈ V → ⦋𝑠 / 𝑡⦌𝐷 = 𝐶)
7	3, 6	ax-mp 5	. . 3 ⊢ ⦋𝑠 / 𝑡⦌𝐷 = 𝐶
8	2, 7	ifbieq2i 4060	. 2 ⊢ if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, ⦋𝑠 / 𝑡⦌𝐷) = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, 𝐶)
9	1, 8	eqtr4i 2635	1 ⊢ 𝑁 = if(𝑠 ≤ (𝑃 ∨ 𝑄), 𝐼, ⦋𝑠 / 𝑡⦌𝐷)

Syntax hints:   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⦋csb 3499  ifcif 4036   class class class wbr 4583  (class class class)co 6549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552

This theorem is referenced by: (None)